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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.stat.inference;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import org.apache.commons.math3.distribution.NormalDistribution;<a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.exception.ConvergenceException;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.exception.MaxCountExceededException;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.exception.NoDataException;<a name="line.22"></a>
<FONT color="green">023</FONT>    import org.apache.commons.math3.exception.NullArgumentException;<a name="line.23"></a>
<FONT color="green">024</FONT>    import org.apache.commons.math3.stat.ranking.NaNStrategy;<a name="line.24"></a>
<FONT color="green">025</FONT>    import org.apache.commons.math3.stat.ranking.NaturalRanking;<a name="line.25"></a>
<FONT color="green">026</FONT>    import org.apache.commons.math3.stat.ranking.TiesStrategy;<a name="line.26"></a>
<FONT color="green">027</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.27"></a>
<FONT color="green">028</FONT>    <a name="line.28"></a>
<FONT color="green">029</FONT>    /**<a name="line.29"></a>
<FONT color="green">030</FONT>     * An implementation of the Mann-Whitney U test (also called Wilcoxon rank-sum test).<a name="line.30"></a>
<FONT color="green">031</FONT>     *<a name="line.31"></a>
<FONT color="green">032</FONT>     * @version $Id: MannWhitneyUTest.java 1416643 2012-12-03 19:37:14Z tn $<a name="line.32"></a>
<FONT color="green">033</FONT>     */<a name="line.33"></a>
<FONT color="green">034</FONT>    public class MannWhitneyUTest {<a name="line.34"></a>
<FONT color="green">035</FONT>    <a name="line.35"></a>
<FONT color="green">036</FONT>        /** Ranking algorithm. */<a name="line.36"></a>
<FONT color="green">037</FONT>        private NaturalRanking naturalRanking;<a name="line.37"></a>
<FONT color="green">038</FONT>    <a name="line.38"></a>
<FONT color="green">039</FONT>        /**<a name="line.39"></a>
<FONT color="green">040</FONT>         * Create a test instance using where NaN's are left in place and ties get<a name="line.40"></a>
<FONT color="green">041</FONT>         * the average of applicable ranks. Use this unless you are very sure of<a name="line.41"></a>
<FONT color="green">042</FONT>         * what you are doing.<a name="line.42"></a>
<FONT color="green">043</FONT>         */<a name="line.43"></a>
<FONT color="green">044</FONT>        public MannWhitneyUTest() {<a name="line.44"></a>
<FONT color="green">045</FONT>            naturalRanking = new NaturalRanking(NaNStrategy.FIXED,<a name="line.45"></a>
<FONT color="green">046</FONT>                    TiesStrategy.AVERAGE);<a name="line.46"></a>
<FONT color="green">047</FONT>        }<a name="line.47"></a>
<FONT color="green">048</FONT>    <a name="line.48"></a>
<FONT color="green">049</FONT>        /**<a name="line.49"></a>
<FONT color="green">050</FONT>         * Create a test instance using the given strategies for NaN's and ties.<a name="line.50"></a>
<FONT color="green">051</FONT>         * Only use this if you are sure of what you are doing.<a name="line.51"></a>
<FONT color="green">052</FONT>         *<a name="line.52"></a>
<FONT color="green">053</FONT>         * @param nanStrategy<a name="line.53"></a>
<FONT color="green">054</FONT>         *            specifies the strategy that should be used for Double.NaN's<a name="line.54"></a>
<FONT color="green">055</FONT>         * @param tiesStrategy<a name="line.55"></a>
<FONT color="green">056</FONT>         *            specifies the strategy that should be used for ties<a name="line.56"></a>
<FONT color="green">057</FONT>         */<a name="line.57"></a>
<FONT color="green">058</FONT>        public MannWhitneyUTest(final NaNStrategy nanStrategy,<a name="line.58"></a>
<FONT color="green">059</FONT>                                final TiesStrategy tiesStrategy) {<a name="line.59"></a>
<FONT color="green">060</FONT>            naturalRanking = new NaturalRanking(nanStrategy, tiesStrategy);<a name="line.60"></a>
<FONT color="green">061</FONT>        }<a name="line.61"></a>
<FONT color="green">062</FONT>    <a name="line.62"></a>
<FONT color="green">063</FONT>        /**<a name="line.63"></a>
<FONT color="green">064</FONT>         * Ensures that the provided arrays fulfills the assumptions.<a name="line.64"></a>
<FONT color="green">065</FONT>         *<a name="line.65"></a>
<FONT color="green">066</FONT>         * @param x first sample<a name="line.66"></a>
<FONT color="green">067</FONT>         * @param y second sample<a name="line.67"></a>
<FONT color="green">068</FONT>         * @throws NullArgumentException if {@code x} or {@code y} are {@code null}.<a name="line.68"></a>
<FONT color="green">069</FONT>         * @throws NoDataException if {@code x} or {@code y} are zero-length.<a name="line.69"></a>
<FONT color="green">070</FONT>         */<a name="line.70"></a>
<FONT color="green">071</FONT>        private void ensureDataConformance(final double[] x, final double[] y)<a name="line.71"></a>
<FONT color="green">072</FONT>            throws NullArgumentException, NoDataException {<a name="line.72"></a>
<FONT color="green">073</FONT>    <a name="line.73"></a>
<FONT color="green">074</FONT>            if (x == null ||<a name="line.74"></a>
<FONT color="green">075</FONT>                y == null) {<a name="line.75"></a>
<FONT color="green">076</FONT>                throw new NullArgumentException();<a name="line.76"></a>
<FONT color="green">077</FONT>            }<a name="line.77"></a>
<FONT color="green">078</FONT>            if (x.length == 0 ||<a name="line.78"></a>
<FONT color="green">079</FONT>                y.length == 0) {<a name="line.79"></a>
<FONT color="green">080</FONT>                throw new NoDataException();<a name="line.80"></a>
<FONT color="green">081</FONT>            }<a name="line.81"></a>
<FONT color="green">082</FONT>        }<a name="line.82"></a>
<FONT color="green">083</FONT>    <a name="line.83"></a>
<FONT color="green">084</FONT>        /** Concatenate the samples into one array.<a name="line.84"></a>
<FONT color="green">085</FONT>         * @param x first sample<a name="line.85"></a>
<FONT color="green">086</FONT>         * @param y second sample<a name="line.86"></a>
<FONT color="green">087</FONT>         * @return concatenated array<a name="line.87"></a>
<FONT color="green">088</FONT>         */<a name="line.88"></a>
<FONT color="green">089</FONT>        private double[] concatenateSamples(final double[] x, final double[] y) {<a name="line.89"></a>
<FONT color="green">090</FONT>            final double[] z = new double[x.length + y.length];<a name="line.90"></a>
<FONT color="green">091</FONT>    <a name="line.91"></a>
<FONT color="green">092</FONT>            System.arraycopy(x, 0, z, 0, x.length);<a name="line.92"></a>
<FONT color="green">093</FONT>            System.arraycopy(y, 0, z, x.length, y.length);<a name="line.93"></a>
<FONT color="green">094</FONT>    <a name="line.94"></a>
<FONT color="green">095</FONT>            return z;<a name="line.95"></a>
<FONT color="green">096</FONT>        }<a name="line.96"></a>
<FONT color="green">097</FONT>    <a name="line.97"></a>
<FONT color="green">098</FONT>        /**<a name="line.98"></a>
<FONT color="green">099</FONT>         * Computes the &lt;a<a name="line.99"></a>
<FONT color="green">100</FONT>         * href="http://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U"&gt; Mann-Whitney<a name="line.100"></a>
<FONT color="green">101</FONT>         * U statistic&lt;/a&gt; comparing mean for two independent samples possibly of<a name="line.101"></a>
<FONT color="green">102</FONT>         * different length.<a name="line.102"></a>
<FONT color="green">103</FONT>         * &lt;p&gt;<a name="line.103"></a>
<FONT color="green">104</FONT>         * This statistic can be used to perform a Mann-Whitney U test evaluating<a name="line.104"></a>
<FONT color="green">105</FONT>         * the null hypothesis that the two independent samples has equal mean.<a name="line.105"></a>
<FONT color="green">106</FONT>         * &lt;/p&gt;<a name="line.106"></a>
<FONT color="green">107</FONT>         * &lt;p&gt;<a name="line.107"></a>
<FONT color="green">108</FONT>         * Let X&lt;sub&gt;i&lt;/sub&gt; denote the i'th individual of the first sample and<a name="line.108"></a>
<FONT color="green">109</FONT>         * Y&lt;sub&gt;j&lt;/sub&gt; the j'th individual in the second sample. Note that the<a name="line.109"></a>
<FONT color="green">110</FONT>         * samples would often have different length.<a name="line.110"></a>
<FONT color="green">111</FONT>         * &lt;/p&gt;<a name="line.111"></a>
<FONT color="green">112</FONT>         * &lt;p&gt;<a name="line.112"></a>
<FONT color="green">113</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;:<a name="line.113"></a>
<FONT color="green">114</FONT>         * &lt;ul&gt;<a name="line.114"></a>
<FONT color="green">115</FONT>         * &lt;li&gt;All observations in the two samples are independent.&lt;/li&gt;<a name="line.115"></a>
<FONT color="green">116</FONT>         * &lt;li&gt;The observations are at least ordinal (continuous are also ordinal).&lt;/li&gt;<a name="line.116"></a>
<FONT color="green">117</FONT>         * &lt;/ul&gt;<a name="line.117"></a>
<FONT color="green">118</FONT>         * &lt;/p&gt;<a name="line.118"></a>
<FONT color="green">119</FONT>         *<a name="line.119"></a>
<FONT color="green">120</FONT>         * @param x the first sample<a name="line.120"></a>
<FONT color="green">121</FONT>         * @param y the second sample<a name="line.121"></a>
<FONT color="green">122</FONT>         * @return Mann-Whitney U statistic (maximum of U&lt;sup&gt;x&lt;/sup&gt; and U&lt;sup&gt;y&lt;/sup&gt;)<a name="line.122"></a>
<FONT color="green">123</FONT>         * @throws NullArgumentException if {@code x} or {@code y} are {@code null}.<a name="line.123"></a>
<FONT color="green">124</FONT>         * @throws NoDataException if {@code x} or {@code y} are zero-length.<a name="line.124"></a>
<FONT color="green">125</FONT>         */<a name="line.125"></a>
<FONT color="green">126</FONT>        public double mannWhitneyU(final double[] x, final double[] y)<a name="line.126"></a>
<FONT color="green">127</FONT>            throws NullArgumentException, NoDataException {<a name="line.127"></a>
<FONT color="green">128</FONT>    <a name="line.128"></a>
<FONT color="green">129</FONT>            ensureDataConformance(x, y);<a name="line.129"></a>
<FONT color="green">130</FONT>    <a name="line.130"></a>
<FONT color="green">131</FONT>            final double[] z = concatenateSamples(x, y);<a name="line.131"></a>
<FONT color="green">132</FONT>            final double[] ranks = naturalRanking.rank(z);<a name="line.132"></a>
<FONT color="green">133</FONT>    <a name="line.133"></a>
<FONT color="green">134</FONT>            double sumRankX = 0;<a name="line.134"></a>
<FONT color="green">135</FONT>    <a name="line.135"></a>
<FONT color="green">136</FONT>            /*<a name="line.136"></a>
<FONT color="green">137</FONT>             * The ranks for x is in the first x.length entries in ranks because x<a name="line.137"></a>
<FONT color="green">138</FONT>             * is in the first x.length entries in z<a name="line.138"></a>
<FONT color="green">139</FONT>             */<a name="line.139"></a>
<FONT color="green">140</FONT>            for (int i = 0; i &lt; x.length; ++i) {<a name="line.140"></a>
<FONT color="green">141</FONT>                sumRankX += ranks[i];<a name="line.141"></a>
<FONT color="green">142</FONT>            }<a name="line.142"></a>
<FONT color="green">143</FONT>    <a name="line.143"></a>
<FONT color="green">144</FONT>            /*<a name="line.144"></a>
<FONT color="green">145</FONT>             * U1 = R1 - (n1 * (n1 + 1)) / 2 where R1 is sum of ranks for sample 1,<a name="line.145"></a>
<FONT color="green">146</FONT>             * e.g. x, n1 is the number of observations in sample 1.<a name="line.146"></a>
<FONT color="green">147</FONT>             */<a name="line.147"></a>
<FONT color="green">148</FONT>            final double U1 = sumRankX - (x.length * (x.length + 1)) / 2;<a name="line.148"></a>
<FONT color="green">149</FONT>    <a name="line.149"></a>
<FONT color="green">150</FONT>            /*<a name="line.150"></a>
<FONT color="green">151</FONT>             * It can be shown that U1 + U2 = n1 * n2<a name="line.151"></a>
<FONT color="green">152</FONT>             */<a name="line.152"></a>
<FONT color="green">153</FONT>            final double U2 = x.length * y.length - U1;<a name="line.153"></a>
<FONT color="green">154</FONT>    <a name="line.154"></a>
<FONT color="green">155</FONT>            return FastMath.max(U1, U2);<a name="line.155"></a>
<FONT color="green">156</FONT>        }<a name="line.156"></a>
<FONT color="green">157</FONT>    <a name="line.157"></a>
<FONT color="green">158</FONT>        /**<a name="line.158"></a>
<FONT color="green">159</FONT>         * @param Umin smallest Mann-Whitney U value<a name="line.159"></a>
<FONT color="green">160</FONT>         * @param n1 number of subjects in first sample<a name="line.160"></a>
<FONT color="green">161</FONT>         * @param n2 number of subjects in second sample<a name="line.161"></a>
<FONT color="green">162</FONT>         * @return two-sided asymptotic p-value<a name="line.162"></a>
<FONT color="green">163</FONT>         * @throws ConvergenceException if the p-value can not be computed<a name="line.163"></a>
<FONT color="green">164</FONT>         * due to a convergence error<a name="line.164"></a>
<FONT color="green">165</FONT>         * @throws MaxCountExceededException if the maximum number of<a name="line.165"></a>
<FONT color="green">166</FONT>         * iterations is exceeded<a name="line.166"></a>
<FONT color="green">167</FONT>         */<a name="line.167"></a>
<FONT color="green">168</FONT>        private double calculateAsymptoticPValue(final double Umin,<a name="line.168"></a>
<FONT color="green">169</FONT>                                                 final int n1,<a name="line.169"></a>
<FONT color="green">170</FONT>                                                 final int n2)<a name="line.170"></a>
<FONT color="green">171</FONT>            throws ConvergenceException, MaxCountExceededException {<a name="line.171"></a>
<FONT color="green">172</FONT>    <a name="line.172"></a>
<FONT color="green">173</FONT>            /* long multiplication to avoid overflow (double not used due to efficiency<a name="line.173"></a>
<FONT color="green">174</FONT>             * and to avoid precision loss)<a name="line.174"></a>
<FONT color="green">175</FONT>             */<a name="line.175"></a>
<FONT color="green">176</FONT>            final long n1n2prod = (long) n1 * n2;<a name="line.176"></a>
<FONT color="green">177</FONT>    <a name="line.177"></a>
<FONT color="green">178</FONT>            // http://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U#Normal_approximation<a name="line.178"></a>
<FONT color="green">179</FONT>            final double EU = n1n2prod / 2.0;<a name="line.179"></a>
<FONT color="green">180</FONT>            final double VarU = n1n2prod * (n1 + n2 + 1) / 12.0;<a name="line.180"></a>
<FONT color="green">181</FONT>    <a name="line.181"></a>
<FONT color="green">182</FONT>            final double z = (Umin - EU) / FastMath.sqrt(VarU);<a name="line.182"></a>
<FONT color="green">183</FONT>    <a name="line.183"></a>
<FONT color="green">184</FONT>            // No try-catch or advertised exception because args are valid<a name="line.184"></a>
<FONT color="green">185</FONT>            final NormalDistribution standardNormal = new NormalDistribution(0, 1);<a name="line.185"></a>
<FONT color="green">186</FONT>    <a name="line.186"></a>
<FONT color="green">187</FONT>            return 2 * standardNormal.cumulativeProbability(z);<a name="line.187"></a>
<FONT color="green">188</FONT>        }<a name="line.188"></a>
<FONT color="green">189</FONT>    <a name="line.189"></a>
<FONT color="green">190</FONT>        /**<a name="line.190"></a>
<FONT color="green">191</FONT>         * Returns the asymptotic &lt;i&gt;observed significance level&lt;/i&gt;, or &lt;a href=<a name="line.191"></a>
<FONT color="green">192</FONT>         * "http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue"&gt;<a name="line.192"></a>
<FONT color="green">193</FONT>         * p-value&lt;/a&gt;, associated with a &lt;a<a name="line.193"></a>
<FONT color="green">194</FONT>         * href="http://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U"&gt; Mann-Whitney<a name="line.194"></a>
<FONT color="green">195</FONT>         * U statistic&lt;/a&gt; comparing mean for two independent samples.<a name="line.195"></a>
<FONT color="green">196</FONT>         * &lt;p&gt;<a name="line.196"></a>
<FONT color="green">197</FONT>         * Let X&lt;sub&gt;i&lt;/sub&gt; denote the i'th individual of the first sample and<a name="line.197"></a>
<FONT color="green">198</FONT>         * Y&lt;sub&gt;j&lt;/sub&gt; the j'th individual in the second sample. Note that the<a name="line.198"></a>
<FONT color="green">199</FONT>         * samples would often have different length.<a name="line.199"></a>
<FONT color="green">200</FONT>         * &lt;/p&gt;<a name="line.200"></a>
<FONT color="green">201</FONT>         * &lt;p&gt;<a name="line.201"></a>
<FONT color="green">202</FONT>         * &lt;strong&gt;Preconditions&lt;/strong&gt;:<a name="line.202"></a>
<FONT color="green">203</FONT>         * &lt;ul&gt;<a name="line.203"></a>
<FONT color="green">204</FONT>         * &lt;li&gt;All observations in the two samples are independent.&lt;/li&gt;<a name="line.204"></a>
<FONT color="green">205</FONT>         * &lt;li&gt;The observations are at least ordinal (continuous are also ordinal).&lt;/li&gt;<a name="line.205"></a>
<FONT color="green">206</FONT>         * &lt;/ul&gt;<a name="line.206"></a>
<FONT color="green">207</FONT>         * &lt;/p&gt;&lt;p&gt;<a name="line.207"></a>
<FONT color="green">208</FONT>         * Ties give rise to biased variance at the moment. See e.g. &lt;a<a name="line.208"></a>
<FONT color="green">209</FONT>         * href="http://mlsc.lboro.ac.uk/resources/statistics/Mannwhitney.pdf"<a name="line.209"></a>
<FONT color="green">210</FONT>         * &gt;http://mlsc.lboro.ac.uk/resources/statistics/Mannwhitney.pdf&lt;/a&gt;.&lt;/p&gt;<a name="line.210"></a>
<FONT color="green">211</FONT>         *<a name="line.211"></a>
<FONT color="green">212</FONT>         * @param x the first sample<a name="line.212"></a>
<FONT color="green">213</FONT>         * @param y the second sample<a name="line.213"></a>
<FONT color="green">214</FONT>         * @return asymptotic p-value<a name="line.214"></a>
<FONT color="green">215</FONT>         * @throws NullArgumentException if {@code x} or {@code y} are {@code null}.<a name="line.215"></a>
<FONT color="green">216</FONT>         * @throws NoDataException if {@code x} or {@code y} are zero-length.<a name="line.216"></a>
<FONT color="green">217</FONT>         * @throws ConvergenceException if the p-value can not be computed due to a<a name="line.217"></a>
<FONT color="green">218</FONT>         * convergence error<a name="line.218"></a>
<FONT color="green">219</FONT>         * @throws MaxCountExceededException if the maximum number of iterations<a name="line.219"></a>
<FONT color="green">220</FONT>         * is exceeded<a name="line.220"></a>
<FONT color="green">221</FONT>         */<a name="line.221"></a>
<FONT color="green">222</FONT>        public double mannWhitneyUTest(final double[] x, final double[] y)<a name="line.222"></a>
<FONT color="green">223</FONT>            throws NullArgumentException, NoDataException,<a name="line.223"></a>
<FONT color="green">224</FONT>            ConvergenceException, MaxCountExceededException {<a name="line.224"></a>
<FONT color="green">225</FONT>    <a name="line.225"></a>
<FONT color="green">226</FONT>            ensureDataConformance(x, y);<a name="line.226"></a>
<FONT color="green">227</FONT>    <a name="line.227"></a>
<FONT color="green">228</FONT>            final double Umax = mannWhitneyU(x, y);<a name="line.228"></a>
<FONT color="green">229</FONT>    <a name="line.229"></a>
<FONT color="green">230</FONT>            /*<a name="line.230"></a>
<FONT color="green">231</FONT>             * It can be shown that U1 + U2 = n1 * n2<a name="line.231"></a>
<FONT color="green">232</FONT>             */<a name="line.232"></a>
<FONT color="green">233</FONT>            final double Umin = x.length * y.length - Umax;<a name="line.233"></a>
<FONT color="green">234</FONT>    <a name="line.234"></a>
<FONT color="green">235</FONT>            return calculateAsymptoticPValue(Umin, x.length, y.length);<a name="line.235"></a>
<FONT color="green">236</FONT>        }<a name="line.236"></a>
<FONT color="green">237</FONT>    <a name="line.237"></a>
<FONT color="green">238</FONT>    }<a name="line.238"></a>




























































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